I'm going to try and play this out:
Akane lived and had never been burned to death. The reports of a kid dying were things past-Akane or present-Akane had told everyone who was there in the Nonary Project on the Gigantic. Whether they were aware that it was a lie or not, the information had to be passed to Junpei, to get him in the state of mind that there was real danger of her/them not living through the incinerator event. That meant:
Snake: Knew that Akane had lived through the event, was in on the lie and told it to Junpei.
Seven: Knew that Akane had lived through the event, was also in on the lie and told it to Junpei. Never had amnesia.
Clover: Did not know that Akane had lived through the event. Thought she had died, and told it to Junpei.
Lotus: Did not know that Akane had lived through the event.
Ace: Knew that Akane had lived through the event. He doesn't reveal this information to keep from revealing his true identity, and Aoi removes him from the incinerator before he ever gets a chance to do so.
And of course, Santa knew she had lived through the event. There were only a couple people who did not know: Clover and Lotus, and it was probably necessary that they didn't know to create a believable story for Junpei.
Really, Lotus and Clover are the two biggest mysteries in the game. Clover I can understand being there, seeing as she was part of the project in that same building, but her being left in the dark is a mystery. Lotus's presence doesn't make much sense to me either. Her only connection was that her daughters had been kidnapped years ago.
But otherwise, Junpei was told exactly what he needed to hear to get the job done.
The only other option is that Akane contacting the future changed the past. That might explain Seven's headache, and if Snake wasn't aware that she was alive when the change to the past had been made, maybe it's why he didn't suffer a headache either (Santa was out of the room, so we couldn't see any sort of reaction). I kinda doubt it though, considering Snake seemed acutely aware of his surroundings and would have known she had made it through the experiment.
Only other question I have is how the hell Akane got out of the incinerator without anyone noticing. Although she was Zero so there was perhaps a chance that she had built some sort of way out.
Also: that was a real-ass incinerator she put everyone in.